The Lattice Field Medium: A Computational Substrate for Emergent Physics

This paper establishes the foundational reference for the Lattice Field Medium (LFM) framework—a computational substrate from which gravitational, electromagnetic, quantum, and cosmological phenomena emerge as effective descriptions.

The Four Canonical Equations (v4.0)

The framework is defined by four canonical equations in a clear hierarchy:

GOV-01 (E Wave Equation) — FUNDAMENTAL
∂²E/∂t² = c²∇²E − χ²E

GOV-02 (χ Wave Equation) — FUNDAMENTAL
∂²χ/∂t² = c²∇²χ − κ(E² − E₀²)

GOV-03 (Fast-Response Simplification)
χ² = χ₀² − g⟨E²⟩_τ

GOV-04 (Poisson Limit) — Quasi-static
∇²χ = (κ/c²)(E² − E₀²)

Equations (1) and (2) form a coupled wave system ensuring strictly causal propagation at speed c with no action-at-a-distance. Gravitational waves emerge as propagating χ-field perturbations.

New Results Since the Previous Version

Background χ normalization.
When the LFM substrate is anchored to relativistic dispersion, QFT mass-shell conventions, and numerically stable lattice evolution, the background stiffness parameter converges to an optimal value χ₀ ≈ 19 at t = 0 (in natural units). This value is not assumed and not a universal constant; it emerges as the unique normalization that simultaneously satisfies relativistic behavior, wave-packet stability, and QFT-consistent unit mapping.

QFT-scale consistency without additional postulates.
Several quantum-field-theory relations appear directly from the LFM dispersion structure once standard unit identifications are applied (ℏ as a conversion factor, not a postulate). In particular:

• the relativistic mass-shell relation E² = p²c² + m²c⁴,

• effective particle masses m = ℏχ/c²,

• and associated relativistic scaling relations
  arise without invoking operator quantization, probability postulates, or external QFT assumptions.

These results follow from wave mechanics on the substrate and do not require separate derivations beyond the governing dispersion relation.

Derivations

From GOV-01, we derive 38 equations classified by epistemic status:
22 exact derivations (DERIVED), 4 limiting cases (LIMIT), 8 phenomenological fits (PHENOM), and 2 requiring external physics (EXTERNAL).

The derivations span wave dynamics, relativistic structure, electromagnetic analogues, quantum bound states, thermodynamic relations, cosmological evolution, and emergent gravity.

Key derived results include:

• Coupling constant γ = 4/3 from the stress-energy tensor

• MOND acceleration scale a₀ = cH₀/(2π) = 1.08 × 10⁻¹⁰ m/s² from cosmic boundary conditions

• Perihelion precession 42.93 arcsec/century (0.12% agreement with GR), resolving the scalar-gravity objection via χ-dependent clocks and rulers

• Dispersion relation ω² = c²k² + χ²

• Gravitational-wave propagation speed v_GW = c

• Frame-dragging emerging from Helmholtz decomposition of momentum flux T₀ᵢ (no vector extension required)

Observational Consistency

The predicted acceleration scale a₀ = cH₀/(2π) was tested against 3,375 SPARC galaxy rotation curves using the standard RAR interpolating function, yielding RMS = 0.024 dex, slightly outperforming empirically calibrated MOND fits (RMS = 0.028 dex).

The BTFR exponent n = 4.29 ± 0.52 at g ≈ a₀ is consistent with LFM predictions in the deep-MOND regime.

Numerical simulations of the coupled GOV-01+GOV-02 system reproduce Keplerian orbits to 0.04% accuracy.

Testable Predictions

The paper consolidates 31 testable predictions across three tiers:

• 7 strong tests (JWST high-z rotation curves, Gaia wide binaries, Rubin stream gaps),

• 9 moderate tests (BTFR evolution, UDG kinematics, lensing–dynamics consistency),

• 5 exploratory tests (CMB lensing, cluster profiles).

Additional consistency checks confirm compatibility with binary pulsars, Solar System dynamics, and the Bullet Cluster.

This document serves as the definitive reference for the LFM framework: what is assumed, what is derived, what emerges from the substrate, and what can be tested experimentally.